Decision making is a process by

which "best solutions‖ are

found.

Managers are the one who make

the decisions to lead to the best

outcome possible under a given

circumstances.

It is an act, process or

methodology of making

something fully perfect,

functional or as effective as

possible.

Case of a MANAGER

- A manager always find the level

of output that maximizes the

profit of the firm or to determine

how much labor, capital and raw

material inputs to use to produce

a given amount of output at the

lowest possible cost.

Case of Consumers

- As consumers, they will search goods

within the constraints imposed by

their prices and their income, for the

combination of goods and services that

will yield the highest level of

satisfaction.

The function the decision maker

seeks to maximize or minimize

Examples:

1. Manager – will always try to

maximize profit.

2. Consumer – will always

maximize consumer goods.

- Optimization problem that involves

maximizing/minimizing the objective

function.

- When the Objective function

measures a benefit, the decision

maker seeks to maximize this benefit

thus solving a maximization problem.

- When the Objective function

measures a cost, the decision maker

seeks to minimize the cost, thus

solving a minimizing problem.

Determines the objective function.

Example:

- Profit

The value of profit will be determined by the

number of units sold or produced while the

production of unit of the good is the activity or

choice variable that determines the value of the

objective function which is profit.

Objective Function- Measures whatever it is

that the particular decision maker wishes to

either maximize or minimize.

E.g. profit, cost, satisfaction…

Maximization Problem- optimizing problem

that involves maximizing the objective

function

Discrete Choice Variable- choice Variable

that can only take a specific integer

Continious Choice Variable- choice variable

that can take on any value between two end

point.

Minimization Problem- optimizing problem

that involves minimizing the objective

function

Activities or Choice variables- Determine the

value of Objective function.

Objective function maybe a function of more

than one activity

Unconstrained optimization- an optimization

problem wherein the decision maker can

choose any level of activity from unrestricted

set of values.

E.g. no external restrictions inchoosing any

level of output in order to maximize net

benefit.

Constrained Optimization- Optimization

problems wherein the decision maker can

choose values for choice variables from a

restricted set of values

Constrained maximization- maximization

problem where activities must be chose to

satisfy a side constraint that the total cost of

activities be held to specific amount

Total benefit function=objective function

Total cost= constraint

Constrained Minimization- minimization

problem where the activities must be chosen

to satisfy a side constraint that the total

benefit of the activities be held to specific

amount.

Objective function= Total cost function

Total benefit function= constraint

E.g. gift shop

Marginal Analysis – analytical tool for solving

optimization problems that involves changing

the value of choice variable by a small

amount to see if the objective function can

be further increased or further decreased

Unconstrained Maximization

NB= TB-TC

NB=net benefit

TB=total benefit

TC=total cost

The activity is increased or decreased in

order to obtain highest level of benefit

The optimal level of activity is obtained

when no further increase in net benefit are

possible in any change of activity

Marginal Benefit- addition to total benefit

attribute to increasing the activity by a small

amount.

Marginal Cost- addition to total cost attribute

to increasing the activity by a small amount

MB= Change in total benefit

Change in activity

MC= Change in total cost

Change in activity

MB>MC MB<MC

Increase activity NB rises NB falls

Decrease activity NB falls NB rises

Optimal level of the activity is attained-net benefit is

maximized-when level of activity is the last level for which

marginal benefit exceeds marginal cost

Maximization with a Continuous Choice variable

When a decision maker wishes to obtain the

maximum net benefit from an activity that is

continuously variable, the optimal level of the

activity is that level at which the marginal

benefit is equal to marginal cost (MB=MC)

Constrained Optimization

- An objective function is maximized or minimized

subject to a constraint if, for all of the activities

in the objective function, the ratios of marginal

benefit per dollar spent be equal for all

activites.

- MBA/PA=MBB/PB

This chapter set forth the basic principles of

regression analysis: estimation and

assessment of statistical significance. We

emphasized how to interpret the results of

regression analysis, rather than focusing on

the mathematics of regression analysis.

The coefficients in an equation that

determine the exact mathematical relation

among variables.

Y being the dependent variable and X the

independent or the explanatory variable.

It is the process of finding estimates of the

numerical values of the parameters of

equation.

The two variable linear model or the simple

regression analyisis is used for testing

hypothesis using the Y variable or the

independent variable and X variable or the

explanatory varible.

YEAR n Yi(Corn) Xi(fertilizer) yi xi xiyi xi2

1971 1 40 6 -17 -12 204 144

1972 2 44 10 -13 -8 104 64

1973 3 46 12 -11 -6 66 36

1974 4 48 14 -9 -4 36 16

1975 5 52 16 -5 -2 10 4

1976 6 58 18 1 0 0 0

1977 7 60 22 3 4 12 16

1978 8 68 24 11 6 66 36

1979 9 74 26 17 8 136 64

1980 10 80 32 23 14 322 196

Total: 10 570 180 0 0 956 576

mean: 57 18

YEAR n Yi(Corn) Xi(fertilizer)

1971 1 40 6

1972 2 44 10

1973 3 46 12

1974 4 48 14

1975 5 52 16

1976 6 58 18

1977 7 60 22

1978 8 68 24

1979 9 74 26

1980 10 80 32

YEAR n Yi(Corn) Xi(fertilizer)

1971 1 40 6

1972 2 44 10

1973 3 46 12

1974 4 48 14

1975 5 52 16

1976 6 58 18

1977 7 60 22

1978 8 68 24

1979 9 74 26

1980 10 80 32

Total: 10 570 180

mean: 57 18

YEAR n Yi(Corn) Xi(fertilizer) yi

1971 1 40 6 -17

1972 2 44 10

1973 3 46 12

1974 4 48 14

1975 5 52 16

1976 6 58 18

1977 7 60 22

1978 8 68 24

1979 9 74 26

1980 10 80 32

Total: 10 570 180

mean: 57 18

YEAR n Yi(Corn) Xi(fertilizer) xi

1971 1 40 6 -12

1972 2 44 10

1973 3 46 12

1974 4 48 14

1975 5 52 16

1976 6 58 18

1977 7 60 22

1978 8 68 24

1979 9 74 26

1980 10 80 32

Total: 10 570 180

mean: 57 18

YEAR n Yi(Corn) Xi(fertilizer) yi xi xiyi

1971 1 40 6 -17 -12 204

1972 2 44 10 -13 -8

1973 3 46 12 -11 -6

1974 4 48 14 -9 -4

1975 5 52 16 -5 -2

1976 6 58 18 1 0

1977 7 60 22 3 4

1978 8 68 24 11 6

1979 9 74 26 17 8

1980 10 80 32 23 14

Total: 10 570 180 0 0

mean: 57 18

YEAR n Yi(Corn) Xi(fertilizer) yi xi xi2

1971 1 40 6 -17 -12 144

1972 2 44 10 -13 -8 64

1973 3 46 12 -11 -6 36

1974 4 48 14 -9 -4 16

1975 5 52 16 -5 -2 4

1976 6 58 18 1 0 0

1977 7 60 22 3 4 16

1978 8 68 24 11 6 36

1979 9 74 26 17 8 64

1980 10 80 32 23 14 196

Total: 10 570 180 0 0 576

mean: 57 18 xi2

b1 (slope of the estimated

regression line) = 1.66

= XiYi / Xi2

b0 (Y intercept) = 27.13

Ŷi (estimated Regression

equation)

= mean of Yi - (bi * mean of

Xi)

= 27.12 + 1.66Xi

0

10

20

30

40

50

60

70

80

90

1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981

Yi

Xi

Coefficients

Standard Error

t Stat P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 27.13 1.979265348

13.70457984

7.74557E-07

22.56080593

31.68919407

22.56080593

31.68919407

X Variable 1

1.66 0.101321087

16.38081745

1.94353E-07

1.426075378

1.893369067

1.426075378

1.893369067

Regression Statistics

Multiple R 0.985418303

R Square 0.971049232

Adjusted R Square 0.967430386

Standard Error 2.431706077

Observations 10

ANOVA

d

f

SS MS F Significance F

Regression 1 1586.694444 1586.694444 268.3311803 1.94353E-07

Residual 8 47.30555556 5.913194444

Total 9 1634

Tcomp is greater than the Tcrit (16.38081745>1.860). Since that is the case then X variable is significant with the margin of error given, which is 5%, to explicate the relationship between X and Y

parameters.

• R2 or the explanatory power of the model is equal to 0.9710 or 97.10%. This explains that fertilizer (X) expresses 97.10% of output change in Corn . The R2 is significantly different from zero.

• In F distribution, the Fcomp explains that the parameters are not all equal to zero. The high value of F ratio implies a significant relationship between the dependent and independent variables.

The test of significance of parameter

estimates passed as well as the test for the

coefficient of multiple determination and

test of the overall significance of the

regression.

Population Regression Line Sample Regression Line

The equation or line

representing the

true or (actual)

relation between

dependent variable

and the explanatory

Variable

The line that best fits

the data in the

sample is call the

sample regression line

An estimator that produces estimates of a

parameter that are on average equal to the

true value of the parameter

The distribution (and relative frequency) of

values b can take because observations on Y

and X come from a random sample

The estimated coefficient is far enough away

from zero

Either sufficiently greater than zero (a

positive estimate) or sufficiently less than

zero (a negative estimate)

t-stat is used to test the hypothesis that the

true value of b equals zero

If the t-stat is greater than the critical value

of t, then the hypothesis that b=0 is rejected

in favor of the alternative hypothesis that

b not =0

When the calculated t-stat exceeds the

critical value of t, b is significantly different

from zero, or equivalently, b is statistically

significant

Using P-value

We use P-value for analyzing data and to make the strongest possible conclusion from the limited data’s that are given.

To get the p-value, you need to have the estimated value then the significance level of the alternative hypothesis then the test statistics.

Decision Criterion for a Hypothesis Test Using the P-value:

If P-value is less than a, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

Examples:

Ha: µ 30 versus Ho: µ = 30

Assumptions: X is normally distributed with s = 8 Test Statistic:

a = .05 RR: z < -1.96 or z >1.96 (P-value < .05)

Calculation: z = 1.54

P-value = 2P(z > |zcalculated|) = 2P(z > |1.54|) = 2P(z < -1.54) = 2(.0618) = .1236

Decision: Fail to reject Ho.

Evaluation of Regression Equation

Regression Equation(y) = a + bx Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2) Intercept(a) = (ΣY - b(ΣX)) / N where x and y are the variables. b = The slope of the regression line a = The intercept point of the regression line and the y axis. N = Number of values or elements X = First Score Y = Second Score ΣXY = Sum of the product of first and Second Scores ΣX = Sum of First Scores ΣY = Sum of Second Scores ΣX2 = Sum of square First Scores Regression Example: To find the Simple/Linear Regression of X Values 60 61 66 63 65

Y Values 3.1 3.2 3.8 4 4.1 To find regression equation, we will first find slope, intercept and use it to form regression equation.. Step 1: Count the number of values. N = 5 Step 2: Find XY, X2

Step 3: Find ΣX, ΣY, ΣXY, ΣX2. ΣX = 311 ΣY = 18.6 ΣXY = 1159.7 ΣX2 = 19359 Step 4: Substitute in the above slope formula given. Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX2 - (ΣX)2) = ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)2) = (5798.5 - 5784.6)/(96795 - 96721) = 13.9/74 = 0.19 Step 5: Now, again substitute in the above intercept formula given. Intercept(a) = (ΣY - b(ΣX)) / N = (18.6 - 0.19(311))/5 = (18.6 - 59.09)/5 = -40.49/5 = -8.098 Step 6: Then substitute these values in regression equation formula Regression Equation(y) = a + bx = -8.098 + 0.19x. Suppose if we want to know the approximate y value for the variable x = 64. Then we can substitute the value in the above equation. Regression Equation(y) = a + bx = -8.098 + 0.19(64). = -8.098 + 12.16 = 4.06

Coefficient of determination

It is used for statistical models whose main purpose is to predict the outcome of the future based by other related information.

Measures percentage variation in Y that can be explained by the X’s through the model Y=Xβ + ε

Proportionate reduction of total variation in Y associated with the use of the set of independent variables X1, X2, …, Xk (assuming a constant term is included in the model)

A goodness of fit measure

Consider Tampa sales example. From printout, R2 = 0.9453. • Interpretation: 94% of the variability observed in sale prices can be explained by assessed values of homes. Thus, the assessed value of the home contributes a lot of information about the home’s sale price. • We can also find the pieces we need to compute R2 by hand in either JMP or SAS outputs: – SSyy is called Sum of Squares of Model in SAS and JMP SSE is called Sum of Squares of Error in both SAS and JMP. • In Tampa sales example, SSyy = 1673142, SSE = 96746 and thus R2 = 1673142 − 96746/1673142 = 0.94.

F-test

F-test is a simultaneous test that if all of the beta=0 (it means that all of

your x’s are useless) or at least one x is not equal to zero (which means

that specific variable is affecting Y).

Ex:. The hypothesis that the means of several normally distributed

populations, all having the same standard deviation, are equal. This is

perhaps the best-known F-test, and plays an important role in the

analysis of variance (ANOVA).

The hypothesis that a data set in a regression analysis follows the

simpler of two proposed linear models that are nested within each other.

Multiple Regression

It’s purpose is to learn more about the relationship between

several independent and dependent variables.

Ex. A car agent having a listing of the following cars in the

following characteristics of that car—transportation, comfort,

style, luxury, fuel economy, etc. Once these following

information has been compiled for the various cars, it will be

interesting to see whether and how these measures relate to

the price for which a car is sold. For example, the space of this

car is better analyst of the price of which a car sells than how

luxurious the car is.

Quadratic Regression Models

Log-Linear Regression Models

Are used when the underlying relation between

X and Y plots as a curve, rather than a

straight line.

An analyst would use nonlinear regression

model when the scatter diagram shows a

curvilinear pattern.

Nonlinear regression is a general technique

to fit a curve through your data.

The purpose of linear regression is to find the

line that comes closest to your data.

Quadratic Regression Model

Log-Linear Regression Model

One of the most useful nonlinear forms for

managerial economics

expressed as Y = a + bX + cX2

Nonlinear model to Linear model

Create a new variable. ―Z‖ defined as Z = X2

Y= a + bX + cZ

Run Regression of Y from X and Z

Y X Z

82 3 9

107 3 9

61 4 16

77 5 25

68 6 36

30 8 64

57 10 100

40 12 144

82 14 196

68 15 225

102 17 289

110 18 324

Dependent Variable: Y F-Ratio: 13.11

Observations: 12

R-

squared: 0.75

Variable

Parameter

Intercept

Standard

Error

T-

Ratio

Intercept 140 17.14 8.17

X -20 4.14 -4.83

Z 1.01 0.5 2

Estimated quadratic regression equation is

Y = 140.08 – 19.51X + 1.01X2

1.01 is also the slope parameter estimate for

X2

The estimated equation can be used to

estimate the value of Y for any particular

value of X

Example: if X = 10

Y will be equal to 45.98

Y=140.08 – 19.51(10) + 1.01(10)2

After which, perform a t-tests to determine

the statistical significance of each

parameters.

Y is related to one or more explanatory variables

in a multiplicative form

Y=aXb Zc

Transform to a linear equation

• Y

bX

Percentage change in

Percentage change in

• Y

cZ

Percentage change in

Percentage change in

Parameters b and c are elasticities.

To transform the equation in to a

linear form, we must use the natural

logarithms of both sides of the

equation.

lnY = (ln a) + b(ln X) + c(ln Z)

If we define: Y’ = a’ + bX’ + cZ’

Example

Variable: Y = aXb

Since Y is positive at all points, parameter a is expected to be positive.

Since Y is decreasing as X increases, the parameter X(b) is expected to be negative.

Y X

2810 20

2825 30

2031 30

2228 40

1620 40

1836 50

1217 60

1110 90

1000 110

420 120

602 140

331 170

To estimate the parameters a and b in a nonlinear equation, we transform the equation by taking logarithms: lnY = ln a + b lnX

LOG Y LOG X

7.94094 2.99573

7.94626 3.4012

7.61628 3.4012

7.70886 3.68888

7.39018 3.68888

7.51534 3.91202

7.10414 4.09434

7.01212 4.49981

6.90776 4.70048

6.04025 4.78749

6.40026 4.94164

5.80212 5.1358

Run Regression

Dependent Variable LOG Y F-Ratio 70

Observations 12 R-Square 0.875

Variable Parameter Intercept

Standard Error

T-Ratio

Intercept 11.06 0.48 23.04

Log X -0.96 0.11 -8.73

To obtain parameter estimates:

note: the slope parameter on Log X is also the exponent on X in the linear equation.

Y=aX(-0.96)

note: since b is an elasticity, 10 percent increase in X results in a 9.6 percent decrease Y.

To obtain the estimate of a:

note: we take the antilog of the estimated value of the intercept

parameter.

= antilog(11.06) = 63, 576

Y=63,576X(-0.96)

Show that the two models are

mathematically equivalent.

logX = 4.5

logY = 6.74 = [11.06 – 0.96(4.5)]

take the antilog of Y and X.

X = 90 Y = 845

Y = 845 = [63,577(90) (-0.96) ]

Regression analysis is simply a tool to provide

the necessary information for a manager to

make decisions that maximizes profits.

It offers managers a way of estimating the

functions they need for managerial decision

making

Reference:

Managerial Economics fifth edition

By Charles Maurice and Christopher R. Thomas